Model-independent determination of the strong phase difference between D ⁰ and D¯ ⁰→ π⁺π⁻π⁺π⁻ amplitudes

For the first time, the strong phase difference between D⁰ and D¯ ⁰→ π⁺π⁻π⁺π⁻ amplitudes is determined in bins of the decay phase space. The measurement uses 818 pb⁻¹ of e⁺e⁻ collision data that is taken at the ψ(3770) resonance and collected by the CLEO-c experiment. The measurement is important for the determination of the CP -violating phase γ in B^± → DK^± (and similar) decays, where the D meson (which represents a superposition of D⁰ and D¯ ⁰) subsequently decays to π⁺π⁻π⁺π⁻. To obtain optimal sensitivity to γ, the phase space of the D → π⁺π⁻π⁺π⁻ decay is divided into bins based on a recent amplitude model of the decay. Although an amplitude model is used to define the bins, the measurements obtained are model-independent. The CP -even fraction of the D → π⁺π⁻π⁺π⁻ decay is determined to be F₊ ^4π = 0.769 ± 0.021 ± 0.010, where the uncertainties are statistical and systematic, respectively. Using simulated B^± → DK^±, D → π⁺π⁻π⁺π⁻ decays, it is estimated that by the end of the current LHC run, the LHCb experiment could determine γ from this decay mode with an uncertainty of (±10 ± 7)°, where the first uncertainty is statistical based on estimated LHCb event yields, and the second is due to the uncertainties on the parameters determined in this paper.